This application claims Paris Convention priority of Japanese Application Nos. 2000-125840 filed Apr. 26, 2000 and 2000-230850 filed Jul. 31, 2000, the complete disclosure of which is(are) hereby incorporated by reference.
1. Field of the Invention
The present invention relates to a method for computer-simulating the shape of the solid-liquid interface between a single crystal and a molten liquid of silicon or the like, said single crystal being pulled by Czochralski (hereinafter referred to as CZ) method, and the distribution of point defects in the single crystal.
2. Description of the Related Art
As a simulation method of this kind, as shown in FIG. 7, there has been known a conventional method which estimates the internal temperature distribution of a silicon molten liquid 2 by operating the thermal conductivity of the molten silicon liquid 2 on the basis of the structure of a hot zone and the pulling speed of a silicon single crystal 4 in a pulling apparatus 1 when pulling the silicon single crystal 4 by means of CZ method using an overall heat transfer simulator and which obtains the shape of the solid-liquid interface between the silicon single crystal 4 and the silicon molten liquid 2 from this internal temperature distribution by means of a computer.
And there has been known another conventional method of obtaining the coordinates and temperature of meshes of a silicon single crystal 4 from the internal temperature distribution of said silicon molten liquid 2 and then solving a diffusion equation on the basis of the diffusion coefficients and the boundary conditions of interstitial silicon atoms and atomic vacancies in the silicon single crystal 4, and thereby obtaining the density distributions of said interstitial silicon atoms and vacancies by means of a computer.
In these methods, each member in the hot zone is mesh-divided and modeled as a mesh structure. Particularly, the silicon molten liquid 2 is divided into comparatively rough meshes of about 10 mm so as to shorten the computation time.
In the above-mentioned conventional methods, however, since the convection of a molten silicon generated in an actual pulling apparatus is not considered and the meshes of the molten silicon are comparatively rough, there has been a problem that a simulation result and an actual measurement result of the shape of a solid-liquid interface are greatly different from each other, and a simulation result (FIG. 6(b)) and an actual measurement result (FIG. 6(e)) of the density distributions of interstitial silicon atoms and vacancies are also greatly different from each other.
An object of the present invention is to provide a method for simulating the shape of the solid-liquid interface between a single crystal and a molten liquid, in which a computation result and an actual measurement result of the shape of the solid-liquid interface between the single crystal and the molten liquid coincide very well with each other.
Another object of the present invention is to provide a method for simulating the distribution of point defects in a single crystal, in which a computation result and an actual measurement result of the distribution of point defects in the single crystal coincide very well with each other.
A first aspect of the present invention is characterized, as shown in FIGS. 1 and 2, by a method for simulating the shape of the solid-liquid interface between a single crystal and a molten liquid using a computer, comprising;
a first step of modeling as a mesh structure a hot zone in a pulling apparatus 11 of the single crystal 14 to be computed,
a second step of combining meshes for each member in the hot zone and inputting physical property values of each member corresponding to the combined meshes into the computer,
a third step of obtaining the surface temperature distribution of each member on the basis of the calorific power of a heater and the emissivity of each member,
a fourth step of obtaining the internal temperature distribution of each member by solving a heat conduction equation on the basis of the surface temperature distribution and the thermal conductivity of each member, and then further obtaining the internal temperature distribution of a molten liquid 12 being in consideration of convection by simultaneously solving a turbulent model equation obtained on the assumption that the molten liquid 12 is a turbulent flow and Navier-Stokes equation, a fifth step of obtaining the shape of the solid-liquid interface between the single crystal 14 and the molten liquid 12 in accordance with an isothermal line including a tri-junction S of the single crystal 14, and
a sixth step of repeating the third to fifth steps until the tri-junction S becomes the melting point of the single crystal 14, wherein;
some or all of the meshes of the molten liquid 12 which are in the radial directions of the single crystal 14 and are directly under the single crystal 14 are set at 0.01 to 5.00 mm, and
some or all of the meshes of the molten liquid 12 which are in the longitudinal direction of the single crystal 14 are set to 0.01 to 5.00 mm.
Since the method for simulating the solid-liquid interface between a single crystal and a molten liquid according to the first aspect of the present invention takes account of convection of the molten liquid 12 and sets comparatively fine meshes of the molten liquid 12, the shape of the solid-liquid interface between the single crystal 14 and the molten liquid 12 obtained by computation coincides very well with an actual measurement result.
It is preferable that the physical property values to be given to each member in the second step are the thermal conductivity, emissivity, viscosity, coefficient of thermal expansion, density and specific heat of each member.
Further, it is preferable that the turbulent model equation is a kl-model equation represented by equation (1), and an optional value within a range of 0.4 to 0.6 is used as a turbulent parameter C of this model equation:                               κ          t                =                              c                          Pr              t                                xc3x97          ρ          xc3x97          C          xc3x97          d          ⁢                      k                                              (        1        )            
Here, xcexat is the turbulent thermal conductivity of a molten liquid, c is the specific heat of the molten liquid, Prt is Prandtl number, xcfx81 is the density of the molten liquid, d is a distance from the inner wall of a crucible containing the molten liquid, and k is the sum square of variable components to the average flow speed of the molten liquid.
As shown in FIGS. 2 to 4, a second aspect of the present invention is a method for simulating the distribution of point defects of a single crystal using a computer, comprising;
a first step of modeling as a mesh structure a hot zone in a pulling apparatus 11 of a single crystal 14 in a state in which the single crystal 14 has been pulled to a specified length by the pulling apparatus 11,
a second step of combining meshes for each member in the hot zone, and inputting the physical property values of each member corresponding to the combined meshes, the pulled length of the single crystal 14 and the pulling speed of the single crystal 14 corresponding to the pulled length into the computer,
a third step of obtaining the surface temperature distribution of each member on the basis of the calorific power of a heater and the emissivity of each member,
a fourth step of obtaining the internal temperature distribution of each member by solving a heat conduction equation on the basis of the surface temperature distribution and the thermal conductivity of each member, and then further obtaining the internal temperature distribution of a molten liquid 12 being in consideration of convection by simultaneously solving a turbulent model equation obtained on the assumption that the molten liquid 12 is a turbulent flow and Navier-Stokes equation,
a fifth step of obtaining the shape of the solid-liquid interface between the single crystal 14 and the molten liquid 12 in accordance with an isothermal line including a tri-junction S of the single crystal 14,
a sixth step of repeating the third to fifth steps until the tri-junction S becomes the melting point of the single crystal 14, computing the temperature distribution inside the pulling apparatus, obtaining the coordinates and temperatures of the meshes of the single crystal 14, and inputting the respective data into the computer,
a seventh step of repeating the first to sixth steps as varying by stages the pulled length of the single crystal 14, computing the temperature distribution in the pulling apparatus 11, obtaining the coordinates and temperatures of the meshes of the single crystal 14, and inputting the respective data into the computer,
an eighth step of inputting the coordinates and temperature data of the meshes of the single crystal 14, and the diffusion coefficients and the boundary conditions of vacancies and interstitial atoms in the single crystal 14 into the computer, and
a ninth step of solving a diffusion equation on the basis of the coordinates and temperatures of the meshes of the single crystal 14 and the diffusion coefficients and boundary conditions of the vacancies and interstitial atoms, and thereby obtaining the density distributions of the vacancies and interstitial atoms after the single crystal 14 has been cooled.
In the method for simulating the distribution of point defects of a single crystal according to the second aspect of the present invention, since the internal temperature distribution of the single crystal 14 is obtained in consideration of convection of the molten liquid 12 and the distribution of point defects in the single crystal 14 is obtained on the basis of this internal temperature distribution of the single crystal 14 and in consideration of the diffusion of point defects in the single crystal 14, the computation values and actual measurement values of the distribution of point defects in the single crystal 14 coincide very well with each other.